变系数分数阶偏微分方程的数值解法

Numerical Solution of Fractional Partial Differential Equations with Variable Coefficients

本文旨在对二维具有变系数的分数阶偏微分方程构造一种有效的有限差分格式,通过对误差的分析得到该差分格式的收敛阶为 ,其中τ表示时间步长,h1,h2表示空间步长,1<α<2。通过能量分析方法证明所提出的有限差分格式的收敛性和稳定性,最后,我们利用高斯迭代法求解给出的数值算例,通过对误差的处理得到上述格式的收敛阶,并通过图像进一步说明该差分格式的有效性,即通过有限差分格式得到的数值解既能保持原问题表现出来的性质,又可以在一定误差允许的范围内满足工程问题的需要。

The purpose of this paper is to construct an efficient finite difference scheme for two-dimensional fractional partial differential equations with variable coefficients. By analyzing the error, the con-vergence order of the difference scheme is , where τ denotes time step, h1,h2 denote space step, 1 . The convergence and stability of the proposed finite difference scheme are proved by energy method. Finally, we use Gauss iterative method to solve the given numerical example, the convergence order of the above scheme is obtained by dealing with the er-ror, and the effectiveness of the scheme is further illustrated by the image, that is, the numerical solution obtained by the finite difference scheme can not only keep the properties of the original problem, but also meet the needs of engineering problems within a certain error allowable range.